Expanded Multipliers

The simple k assumes no government or overseas (o/s) sectors in the economy. In reality Y will be subject to taxation, imports and savings before it will be spent in the domestic economy. Thus various complex multipliers incorporates these flows and is a smaller coefficient than [k = 1/MPS].

b - Marginal Propensity to Consume (MPC)
t - Marginal Propensity to Tax (MPT)
m - Marginal Propensity to Import (MPM)

The Multiplier

${1 \over 1-[b\cdot (1-t)-m]}\ \ \ i.e\ \ {1 \over 1-[0.7(1-0.1)-0.2]}={1 \over 0.57}\approx 1.75$

The operation of the multiplier is essentially the same but consumption is reduced by acknowledging the other leakages in the economy.
This multiplier is applicable to all direct spending in the economy, including government spending.

Taxes/Transfers Multiplier

${-b \over 1-[b\cdot (1-t)-m]}$

This multiplier is applicable to all indirect injections into the economy, such as cutting taxes or increasing government transfers. The multiplier is positive because it implies a positive change, or increase, in the input. (taxes).

Balanced Budget Multiplier

${1-b \over 1-[b\cdot (1-t)-m]}$

This multiplier is used when the government increases spending and uses a tax increase to pay for the spending. To model change in expenditure when the government decreases spending and cuts taxes to cover the costs, multiply by a negative input in expenditure.

As you can see, the result is still positive if you increase both government spending and taxes. On the other hand, if you decrease both government spending and taxes, you have a negative change in expenditures, i.e. widening the recession gap.

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